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Additional resources for Applied Factor Analysis in the Natural Sciences

Example text

6, we explained that the product of two conformable matrices A ( 5 X r ) and B (rXr) is a matrix C of order s by t. We now consider the reverse process of factoring a product into its components. In scalar algebra, any number can be decomposed into factors, usually in an infinite number of ways. For example, the number 12 can be considered as the product of the factors 3 and 4, or 2 and 6, or 1 and 12, and so on. Similarly, not only can any matrix be thought of as the pfoduct of two factors but also there is an infinite number of such pairs of factors (Horst, 1963).

In Fig. 16A, the three column vectors are embedded in the three-dimensional object space. Because there are no more than two linearly independent vectors in the set, the vectors are located in a two-dimensional subspace. The two linearly independent vectors are said to form a basis for this subspace. For the general case, consider a data matrix X(NXp) where only r of the p column vectors are linearly independent. Then the p column vectors will span the r-dimensional subspace of the original Af-space, and the r linearly independent vectors will form a basis of this subspace.

However, in a similar fashion one can proceed to consider the regression of y 2 on all the other variables (including yx), of y 3 on all the other variables, and so on. In compact form we can now write y = By + u where y is the vector of all p variables, u is the vector of all p residuals, and B is a matrix whose diagonal elements are zero (since y i is not involved on the right side of the /th equation) and the off-diagonal elements are the regression coefficients. Then B = I - (diagS"1)"^ and the residual variances are given by the reciprocals of the diagonal elements of S" 1 .